Definition
Let be a Lie algebra over a commutative ring R with universal enveloping algebra, and let M be a representation of (equivalently, a -module). Considering R as a trivial representation of, one defines the cohomology groups
(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor
Analogously, one can define Lie algebra homology as
(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor
Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.
Read more about this topic: Lie Algebra Cohomology
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